I'd like to answer just the question title, with a suggestion, based on the kinds of rule tweaking I often employ in D&D:
How should I modify this crouching rule so that it grants some sort of mechanical advantage, but not big enough so players can abuse it?
Going into a crouch, and coming out of a crouch, costs 5 feet of movement. Movement whilst crouching costs +1 foot per foot of distance. This is strictly better than dropping prone and getting back up on their turn.
A crouching character upgrades their cover by a single step against Ranged and Melee attacks (i.e. the usual cover bonus appliesthis does not improve dexterity saves against area effect spells). If they upgrade from 3/4 to total cover, then they can no longer see their target either - e.g. think ducking or moving away when behind an arrow slit. This is usually a worse benefit than dropping prone, or could be equally achieved by moving around a corner etc.
Melee attacks from adjacent positions against crouching characters gain Advantage, same as against prone characters. Note this is somewhat countered by the additional cover that the character has gained, but as an attacker, I'd usually still opt for -2 with Advantage.
I have not played those exact rules above, but have ruled similarly in ad-hoc situations where there was certain types of cover that players wanted to take better advantage of - e.g. gravestones in a graveyard. It worked fine in those circumstances, and gave a feeling of using suitable tactics in a ranged battle.